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Kenichiro Tanabe

Publications and source records attributed to Kenichiro Tanabe.

17 recordsLinked to original sources
arXiv2023

A Schur-Weyl type duality for twisted weak modules over a vertex algebra

Let $V$ be a vertex algebra of countable dimension, $G$ a subgroup of ${\rm Aut} V$ of finite order, $V^{G}$ the fixed point subalgebra of $V$ under the action of $G$, and ${\mathscr S}$ a finite $G$-stable set of inequivalent irreducible twisted weak $V$-modules associated with possibly different automorphisms in $G$. We show a Schur--Weyl type duality for the actions of ${\mathscr A}_α(G,{\mathscr S})$ and $V^G$ on the direct sum of twisted weak $V$-modules in ${\mathscr S}$ where ${\mathscr A}_α(G,{\mathscr S})$ is a finite dimensional semisimple associative algebra associated with $G,{\mathscr S}$, and a $2$-cocycle $α$ naturally determined by the $G$-action on ${\mathscr S}$. It follows as a natural consequence of the result that for any $g\in G$ every irreducible $g$-twisted weak $V$-module is a completely reducible weak $V^G$-module.

math.QA↗
arXiv2023

The irreducible weak modules for the fixed point subalgebra of the vertex algebra associated to a non-degenerate even lattice by an automorphism of order $2$ (Part $2$)

Let $V_{L}$ be the vertex algebra associated to a non-degenerate even lattice $L$, $θ$ the automorphism of $V_{L}$ induced from the $-1$ symmetry of $L$, and $V_{L}^{+}$ the fixed point subalgebra of $V_{L}$ under the action of $θ$. In this series of papers, we classify the irreducible weak $V_{L}^{+}$-modules and show that any irreducible weak $V_{L}^{+}$-module is isomorphic to a weak submodule of some irreducible weak $V_{L}$-module or to a submodule of some irreducible $θ$-twisted $V_{L}$-module. Let $M(1)^{+}$ be the fixed point subalgebra of the Heisenberg vertex operator algebra $M(1)$ under the action of $θ$. In this paper (Part $2$), we show that there exists an irreducible $M(1)^{+}$-submodule in any non-zero weak $V_{L}^{+}$-module and we compute extension groups for $M(1)^{+}$.

math.QA↗
arXiv2021

The irreducible weak modules for the fixed point subalgebra of the vertex algebra associated to a non-degenerate even lattice by an automorphism of order $2$ (Part $1$)

Let $V_{L}$ be the vertex algebra associated to a non-degenerate even lattice $L$, $θ$ the automorphism of $V_{L}$ induced from the $-1$-isometry of $L$, and $V_{L}^{+}$ the fixed point subalgebra of $V_{L}$ under the action of $θ$. In this series of papers, we classify the irreducible weak $V_{L}^{+}$-modules and show that any irreducible weak $V_{L}^{+}$-module is isomorphic to a weak submodule of some irreducible weak $V_{L}$-module or to a submodule of some irreducible $θ$-twisted $V_{L}$-module. In this paper (Part 1), we show that when the rank of $L$ is $1$, every non-zero weak $V_{L}^{+}$-module contains a non-zero $M(1)^{+}$-module, where $M(1)^{+}$ is the fixed point subalgebra of the Heisenberg vertex operator algebra $M(1)$ under the action of $θ$.

math.QA↗
arXiv2020

The irreducible weak modules for the fixed point subalgebra of the vertex algebra associated to a non-degenerate even lattice by an automorphism of order $2$

Let $V_{L}$ be the vertex algebra associated to a non-degenerate even lattice $L$, $θ$ the automorphism of $V_{L}$ induced from the $-1$ symmetry of $L$, and $V_{L}^{+}$ the fixed point subalgebra of $V_{L}$ under the action of $θ$. We classify the irreducible weak $V_{L}^{+}$-modules and show that any irreducible weak $V_{L}^{+}$-module is isomorphic to a weak submodule of some irreducible weak $V_{L}$-module or to a submodule of some irreducible $θ$-twisted $V_{L}$-module.

math.QA↗
arXiv2017

A generalization of intertwining operators for vertex operator algebras

We generalize the notion of an intertwining operator to N-graded weak modules over a vertex operator algebra and study their properties. We show a formula for the dimensions of these intertwining operators in terms of modules over the Zhu algebras under some conditions on N-graded weak modules.

math.QA↗
arXiv2017

Simple weak modules for the fixed point subalgebra of the Heisenberg vertex operator algebra of rank $1$ by an automorphism of order $2$ and Whittaker vectors

Let $M(1)$ be the vertex operator algebra with the Virasoro element $ω$ associated to the Heisenberg algebra of rank $1$ and let $M(1)^{+}$ be the subalgebra of $M(1)$ consisting of the fixed points of an automorphism of $M(1)$ of order $2$. We classify the simple weak $M(1)^{+}$-modules with a non-zero element $w$ such that for some integer $s\geq 2$, $ω_i w\in{\mathbb C}w$ ($i=\lfloor s/2\rfloor+1,\lfloor s/2\rfloor+2,\ldots,s-1$), $ω_{s}w\in{\mathbb C}^{\times}w$, and $ω_i w=0$ for all $i>s$. The result says that any such simple weak $M(1)^{+}$-module is isomorphic to some simple weak $M(1)$-module or to some $θ$-twisted simple weak $M(1)$-module.

math.QA↗
arXiv2017

Simple weak modules for some subalgebras of the Heisenberg vertex algebra and Whittaker vectors

Let ${\mathscr M}(p)$ $(p=2,3,\ldots)$ be the singlet vertex operator algebra and $ω$ its conformal vector. We classify the simple weak ${\mathscr M}(p)$-modules with a non-zero element $u$ such that for some integer $s\geq 2$, $ω_i u\in{\mathbb C} u$ ($i=\lfloor s/2\rfloor+1,\lfloor s/2\rfloor+2,\ldots,s-1$), $ω_{s} u\in{\mathbb C}^{\times} u$, and $ω_i u=0$ for all $i>s$.

math.QA↗
arXiv2013

A generalization of twisted modules over vertex algebras

We introduce a notion of a (V,T)-module over a vertex algebra V for an arbitrary positive integer T, which is a generalization of a twisted V-module. Under some conditions on V, we construct an associative algebra A^{T}_{m}(V) for m\in(1/T)\N and an A^{T}_{m}(V)-A^{T}_{n}(V)-bimodule A^{T}_{n,m}(V) for n,m\in(1/T)\N and we establish a one-to-one correspondence between the set of isomorphism classes of simple left A^{T}_{0}(V)-modules and that of simple (1/T)\N-graded (V,T)-modules.

math.QA↗
arXiv2013

Fixed point subalgebras of lattice vertex operator algebras by an automorphism of order three

We study the fixed point subalgebra of a certain class of lattice vertex operator algebras by an automorphism of order 3, which is a lift of a fixed-point-free isometry of the underlying lattice. We classify the irreducible modules for the subalgebra. Moreover, the rationality and the $C_2$-cofiniteness of the subalgebra are established. Our result contains the case of the vertex operator algebra associated with the Leech lattice.

math.QA↗
arXiv2011

Finite-dimensional vertex algebra modules over fixed point differential subfields

Let $K$ be a differential field over $\C$ with derivation $D$, $G$ a finite linear automorphism group over $K$ which preserves $D$, and $K^G$ the fixed point subfield of $K$ under the action of $G$. We show that every finite-dimensional vertex algebra $K^G$-module is contained in some twisted vertex algebra $K$-module.

math.QA↗
arXiv2010

Finite-dimensional vertex algebra modules over fixed point commutative subalgebras

Let $A$ be a connected commutative $\C$-algebra with derivation $D$, $G$ a finite linear automorphism group of $A$ which preserves $D$, and $R=A^G$ the fixed point subalgebra of $A$ under the action of $G$. We show that if $A$ is generated by a single element as an $R$-algebra and is a Galois extension over $R$ in the sense of M. Auslander and O. Goldman, then every finite-dimensional vertex algebra $R$-module has a structure of twisted vertex algebra $A$-module.

math.QA↗
arXiv2007

Finite-dimensional modules for the polynomial ring in one variable as a vertex algebra

A commutative associative algebra $A$ over ${\mathbb C}$ with a derivation is one of the simplest examples of a vertex algebra. However, the differences between the modules for $A$ as a vertex algebra and the modules for $A$ as an associative algebra are not well understood. In this paper, I give the classification of finite-dimensional indecomposable untwisted or twisted modules for the polynomial ring in one variable over ${\mathbb C}$ as a vertex algebra.

math.QA↗
arXiv2004

Some algebra related to $P$-and $Q$-polynomial association schemes

Let $K$ denote a field, and let $V$ denote a vector space over $K$ with finite positive dimension. Consider a pair of linear transformations $A:V\to V$ and $A^*:V\to V$ that satisfy both conditions below: (i) There exists a basis for $V$ with respect to which the matrix representing $A$ is diagonal, and the matrix representing $A^*$ is irreducible tridiagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A^*$ is diagonal, and the matrix representing $A$ is irreducible tridiagonal. Such a pair is called a Leonard pair on $V$. In this paper we introduce a mild generalization of a Leonard pair called a tridiagonal pair. A Leonard pair is the same thing as a tridiagonal pair such that for each transformation all eigenspaces have dimension one.

math.CO↗
arXiv2002

Uniform product of A_{g,n}(V) for an orbifold model V and G-twisted Zhu algebra

Let V be a vertex operator algebra and G a finite automorphism group of V. For each g\in G and nonnegative rational number n\in {\mathbb Z}/|g|, a g-twisted Zhu algebra A_{g,n}(V) plays an important role in the theory of vertex operator algebras, but the given product in A_{g,n}(V) depends on the eigenspaces of g. We show that there is a uniform definition of products on V and we introduce a G-twisted Zhu algebra A_{G,n}(V) which covers all g-twisted Zhu algebras. Assume that V is simple and let {\cal S} be a finite set of inequivalent irreducible twisted V-modules which is closed under the action of G. There is a finite dimensional semisimple associative algebra {\cal A}_α(G,{\cal S}) for a suitable 2-cocycle naturally determined by the G-action on {\cal S}. We show that a duality theorem of Schur-Weyl type holds for the actions of {\cal A}_α(G,{\cal S}) and V^G on the direct sum of twisted V-modules in {\cal S} as an application of the theory of A_{G,n}(V). It follows as a natural consequence of the result that for any g\in G every irreducible g-twisted V-module is a completely reducible V^G-module.

math.QA↗